Example 8.4.  Find the residue of  [Graphics:Images/ResidueCalcMod_gr_84.gif]  at  [Graphics:Images/ResidueCalcMod_gr_85.gif].  

Explore Solution 8.4.

Enter the function  [Graphics:../Images/ResidueCalcMod_gr_93.gif].   

[Graphics:../Images/ResidueCalcMod_gr_94.gif]



[Graphics:../Images/ResidueCalcMod_gr_95.gif]

 

 

The function  f[z]  has a pole of order  3  at  [Graphics:../Images/ResidueCalcMod_gr_96.gif].  The residue is computed using Theorem 8.2 (c).

[Graphics:../Images/ResidueCalcMod_gr_97.gif]



[Graphics:../Images/ResidueCalcMod_gr_98.gif]

 

 

Aside. We can verify this computation with Mathematica's built in "Residue" procedure.

[Graphics:../Images/ResidueCalcMod_gr_99.gif]



[Graphics:../Images/ResidueCalcMod_gr_100.gif]

 

 

Which is the coefficient of  [Graphics:../Images/ResidueCalcMod_gr_101.gif] in the Laurent series expansion for  f[z].

[Graphics:../Images/ResidueCalcMod_gr_102.gif]



[Graphics:../Images/ResidueCalcMod_gr_103.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell